While most of teaching effectiveness comes from mastery of basic practices - conveying an enthusiasm for the subject, presenting material in a clear and organized form, and establishing communication and trust with the students - the teachers and mentors who have influenced my teaching the most were also able to anticipate the major difficulties for their students and act accordingly. In teaching across the university mathematics curriculum, difficulties I have found to be most prominent are those which students have in:
By far the most difficult transition a student faces in his or her mathematics training is that of doing proofs. I have tried my hand at designing a bridge course whose structure facilitates that transition. This structure is one of a few different inquiry proof teaching techniques which are being developed by different groups of faculty.
To foster independence, at the advanced undergraduate and graduate levels, I sometimes assign short problem sets which ask questions which students should be asking themselves. These questions are most often elaborations of the question "how does this work in some example?"
I have also encouraged development of basic skills of students outside the college setting. For three years while I was living in Providence I served on the executive committee of the board of directors for the Mt. Hope Learning Center, a community-based educational center located in a disadvantaged neighborhood near Brown University. I am now deeply involved in supporting implementation of the Common Core.
Here is site which usefully links to many leading mathematical organizations and mathematics teaching sites.